#10412124

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#54197	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,8)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(17,18,23,26)(19,24,30,32)(20,25,28,29)(21,31,27,22)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(2,5)(8,10)(9,13)(15,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,32)(27,28);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,20)(18,25)(19,27)(21,30)(22,24)(23,28)(26,29)(31,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,20)(18,25)(19,21)(22,32)(23,28)(24,31)(26,29)(27,30)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,21)(18,22)(19,28)(20,30)(23,27)(24,25)(26,31)(29,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22)(18,21)(19,29)(20,24)(23,31)(25,30)(26,27)(28,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319320);